# CMPSCI 501: Theory of Computation

### David Mix Barrington

### Spring, 2015

# Homework Assignment #5

#### Posted Monday 6 April 2015

#### Due on paper in class, Friday day 17 April 2015

There are fourteen
questions for 100 total points plus
10
extra credit.
All are from
the textbook, *Introduction to the Theory of Computation*
by Michael Sipser (second/third edition). though some are adapted as
indicated.
The number in parentheses following each problem
is its individual point value.

Students are responsible for understanding and following
the academic honesty
policies indicated on this page.

- Problem 6.27 (3rd) (not in 2nd) (10): Let S = {(M): M is a TM and
L(M) = {(M)} (that is, S is the set of TM's that accept their own
description and nothing else). Show that neither S nor its complement
is Turing recognizable.
- Exercise 7.4 (5):
- Exercise 7.6 (5):
- Exercise 7.10 (5):
- Exercise 7.12 (3rd) (not in 2nd) (5):
Call graphs G and H
**isomorphic**
if the nodes of G may be reordered so that it is identical to H. Let
ISO = {(G, H): G and H are isomorphic graphs}. Show that ISO ∈ NP.
- Problem 7.13 (2nd)/7.14 (3rd) (10):
- Problem 7.14 (2nd)/7.15 (3rd) (5):
- Problem 7.17 (2nd)/7.18 (3rd) (5):
- Problem 7.24 (2nd)/7.26 (3rd) (10):
- Problem 7.25 (2nd)/7.27 (3rd) (10):
- Problem 7.35 (2nd)/7.36 (3rd) (10XC):
- Problem 7.44 (2nd)/7.46 (3rd) (10):
- Problem 7.54 (3rd) (not in 2nd) (10):
In a directed graph, the
**indegree** of a node is the number of incoming
edges and the **outdegree** is the number of outgoing edges. Show that the
following problem is NP-complete. Given an undirected graph G and a
designated subset C of G's nodes, is it possible to convert G to a directed
graph by assigning directions to each of its edges so that every node in C
has indegree 0 or outdegree 0, and every other node in G has indegree at least
1?
- Problem 8.9 (10):

Last modified 6 April 2015